1. Consider the system
x′(t) = ax - bxy
y′(t) = cxy - dy
This is known as the Lotka-Volterra predator-prey model for two populations with x(t) being the number of prey and y(t) the number of predators at time t.
(a) Leta=4,b=2,c=1,d=3andsolvethemodelfor0≤t≤5. The initial values are x(0) = 3, y(0) = 5. Plot x and y as functions of t and plot x versus y.
(b) Solve the same model with x(0) = 3 and, in succession, y(0) = .5, 1, 1.5, 2. Plot x versus y in each case? What do you observe? Why would the point (3, 2) be an equilibrium point?
To solve the above system you are to use the fourth order Adams-Bashforth Adams-Moulton Predictor corrector method using the fourth order Runge-Kutta to start the method.
Use a time step of .005.
2 (a). Find the general solution of y′′(x) + 1006y′(x) + 1005y(x) = 0.
by methods you learned in differential equations. Impose the initial conditions y(0) = 1 and y′(0) = -1, what is the solution with these conditions?
(b). Write the above second order equation as a system of first order equations.
(c). Solve the system numerically from x = 0 to x = 2 using the Runge-Kutta method of fourth order for h = .001, .002, .004, .01. For each h print out 3 columns; x, the exact solution at x, and the numerical approximation. Print the values at the equally spaced points: x = 0, .4, .8, 1.2, 1.6, 2. Print to about machine precision.
(d). The numerical approximation should explode at certain values of h. Try to find an approximation to this value of h numerically. Experiment with your working program. Why and when does the happen? How is it explained by the theory?